Sub-Lorentzian Heisenberg Group

• Sub-Lorentzian Heisenberg group is defined by equipping the first Heisenberg group with a Lorentzian metric on its horizontal distribution, combining control theory, causal geometry, and nonholonomy.
• The model offers explicit maximizing geodesics and a globally optimal synthesis derived through both Hamiltonian and variational methods, revealing unique time-separation properties.
• It exhibits a Lorentzian Hausdorff dimension of 4 and demonstrates a sharp failure of synthetic curvature and measure contraction conditions, driving new research in optimal transport and CMC surfaces.

The sub-Lorentzian Heisenberg group is the first Heisenberg group $H^1$ equipped with a Lorentzian metric on its horizontal distribution rather than on the full tangent bundle. In the recent literature, this structure is treated simultaneously as a control system, a Lorentzian length space, a metric-spacetime model for optimal transport, and a test case for synthetic curvature and variational problems. The resulting geometry combines Hörmander-type nonholonomy with causal notions such as timelike, null, and future-directed horizontality; it admits explicit maximizing geodesics, a global optimal synthesis on the timelike reachable set, Lorentzian Hausdorff dimension $4$, and a sharp failure of known timelike synthetic curvature-dimension and measure-contraction conditions. It also supports a developing theory of Lorentzian optimal transport and constant horizontal mean curvature surfaces (Borza et al., 8 Sep 2025, Borza et al., 3 Apr 2025, Sachkov et al., 2022, Borza et al., 29 May 2026).

1. Lie-theoretic model and horizontal Lorentzian structure

The underlying manifold is $\mathbb{R}^3$, endowed with the non-abelian group law

$$(x,y,z) * (x',y',z') = \bigl(x+x',\, y+y',\, z+z' + \tfrac12(x y' - y x')\bigr).$$

The identity is $e=(0,0,0)$, inverses are $(x,y,z)^{-1}=(-x,-y,-z)$, and left translations are smooth automorphisms. In exponential coordinates, the standard left-invariant frame is

$$X=\partial_x-\frac{y}{2}\partial_z,\qquad Y=\partial_y+\frac{x}{2}\partial_z,\qquad T=\partial_z,$$

with

$[X,Y]=T.$

Accordingly, the horizontal distribution is

$D=\mathrm{span}\{X,Y\}\subset T(H^1),$

or equivalently $\mathcal H=\ker\theta$ for the standard contact form

$4$0

Because $4$1, the distribution is bracket-generating but non-involutive, so Hörmander’s condition holds. Heisenberg dilations are

$4$2

These basic constructions are shared across the contemporary treatments of the subject (Borza et al., 8 Sep 2025, Borza et al., 3 Apr 2025, Borza et al., 29 May 2026).

A sub-Lorentzian structure is then obtained by placing a Lorentzian metric on $4$3 with signature $4$4, normalized by

$4$5

Thus $4$6 is timelike, $4$7 is spacelike, and $4$8 is a Lorentz-orthonormal basis of the horizontal bundle. A horizontal vector $4$9 is timelike if $\mathbb{R}^3$0, null if $\mathbb{R}^3$1 and $\mathbb{R}^3$2, and spacelike if $\mathbb{R}^3$3 or $\mathbb{R}^3$4. Future direction is fixed by $\mathbb{R}^3$5, equivalently by $\mathbb{R}^3$6. In control form, horizontal curves satisfy

$\mathbb{R}^3$7

and the induced squared norm is

$\mathbb{R}^3$8

This framework distinguishes the sub-Lorentzian Heisenberg group from full Lorentzian problems on the Heisenberg group. Some papers study left-invariant Lorentzian metrics on all three directions $\mathbb{R}^3$9, whereas the sub-Lorentzian model restricts the metric to $$(x,y,z) * (x',y',z') = \bigl(x+x',\, y+y',\, z+z' + \tfrac12(x y' - y x')\bigr).$$0 and treats the vertical direction as generated only through the bracket (Sachkov, 24 Nov 2025, Galyaev et al., 2024).

2. Causality, proper time, and time-separation

A horizontal curve is causal, timelike, null, or future-directed when its almost-everywhere velocity has the corresponding causal type. For a future-directed causal horizontal curve $$(x,y,z) * (x',y',z') = \bigl(x+x',\, y+y',\, z+z' + \tfrac12(x y' - y x')\bigr).$$1, the Lorentzian length, or proper time, is

$$(x,y,z) * (x',y',z') = \bigl(x+x',\, y+y',\, z+z' + \tfrac12(x y' - y x')\bigr).$$2

and in control variables this becomes

$$(x,y,z) * (x',y',z') = \bigl(x+x',\, y+y',\, z+z' + \tfrac12(x y' - y x')\bigr).$$3

with admissibility condition $$(x,y,z) * (x',y',z') = \bigl(x+x',\, y+y',\, z+z' + \tfrac12(x y' - y x')\bigr).$$4 in the future-directed nonspacelike cone. The time-separation function $$(x,y,z) * (x',y',z') = \bigl(x+x',\, y+y',\, z+z' + \tfrac12(x y' - y x')\bigr).$$5 is defined by

$$(x,y,z) * (x',y',z') = \bigl(x+x',\, y+y',\, z+z' + \tfrac12(x y' - y x')\bigr).$$6

when such curves exist, and $$(x,y,z) * (x',y',z') = \bigl(x+x',\, y+y',\, z+z' + \tfrac12(x y' - y x')\bigr).$$7 otherwise. It satisfies the reverse triangle inequality: $$(x,y,z) * (x',y',z') = \bigl(x+x',\, y+y',\, z+z' + \tfrac12(x y' - y x')\bigr).$$8 Left translations preserve causality, future direction, $$(x,y,z) * (x',y',z') = \bigl(x+x',\, y+y',\, z+z' + \tfrac12(x y' - y x')\bigr).$$9, and $e=(0,0,0)$0 (Borza et al., 8 Sep 2025, Borza et al., 3 Apr 2025).

Horizontality imposes the area-lift relation

$e=(0,0,0)$1

hence

$e=(0,0,0)$2

The vertical coordinate therefore records the signed area swept by the planar projection. This observation is central both in the variational analysis of geodesics and in the optimal transport theory.

Recent papers give explicit descriptions of the future and chronological future of the origin, but not always with the same vertical-sign convention. One formulation gives

$e=(0,0,0)$3

while other treatments record

$e=(0,0,0)$4

This suggests differing conventions in the literature for the vertical coordinate or for the future/past description (Borza et al., 8 Sep 2025, Sachkov et al., 2022, Borza et al., 3 Apr 2025).

The time-separation is continuous on the causal relation set and real-analytic on the chronological relation set. An explicit formula is available on $e=(0,0,0)$5: if $e=(0,0,0)$6 is the inverse of

$e=(0,0,0)$7

then

$e=(0,0,0)$8

In particular, when the endpoints have the same $e=(0,0,0)$9-coordinate,

$(x,y,z)^{-1}=(-x,-y,-z)$0

and always

$(x,y,z)^{-1}=(-x,-y,-z)$1

The same formula underlies the explicit sub-Lorentzian distance and the description of spheres in the optimal-synthesis approach (Borza et al., 3 Apr 2025, Sachkov et al., 2022).

3. Maximizing geodesics and global optimal synthesis

The modern description of maximizing geodesics rests on two complementary viewpoints. The first is Hamiltonian, via Pontryagin’s Maximum Principle. The controlled Hamiltonian is

$(x,y,z)^{-1}=(-x,-y,-z)$2

and the maximized Hamiltonian is

$(x,y,z)^{-1}=(-x,-y,-z)$3

Its Hamilton equations are

$(x,y,z)^{-1}=(-x,-y,-z)$4

For strictly normal timelike extremals, one obtains explicit hyperbolic solutions. At the origin, with initial covector $(x,y,z)^{-1}=(-x,-y,-z)$5,

$(x,y,z)^{-1}=(-x,-y,-z)$6

$(x,y,z)^{-1}=(-x,-y,-z)$7

with continuous interpretation as $(x,y,z)^{-1}=(-x,-y,-z)$8. Equivalent formulas appear in older normal-geodesic calculations and in the later exponential-map description (Borza et al., 3 Apr 2025, Korolko et al., 2010, Sachkov et al., 2022).

The second viewpoint is variational. Projecting to the Minkowski plane $(x,y,z)^{-1}=(-x,-y,-z)$9, every horizontal curve $$X=\partial_x-\frac{y}{2}\partial_z,\qquad Y=\partial_y+\frac{x}{2}\partial_z,\qquad T=\partial_z,$$0 satisfies

$$X=\partial_x-\frac{y}{2}\partial_z,\qquad Y=\partial_y+\frac{x}{2}\partial_z,\qquad T=\partial_z,$$1

while the vertical coordinate fixes the signed area enclosed by the planar projection and its chord. Maximizing geodesics in $$X=\partial_x-\frac{y}{2}\partial_z,\qquad Y=\partial_y+\frac{x}{2}\partial_z,\qquad T=\partial_z,$$2 therefore correspond bijectively to future-directed causal curves in the Minkowski plane that maximize Minkowski proper time subject to a fixed signed-area constraint. The planar isoperimetric problem has three regimes: the maximizer is a straight timelike segment when the prescribed area is zero, a broken null line with one breakpoint at the maximal-area threshold, and an arc of a timelike hyperbola in the strictly interior regime. This gives a direct explanation of why maximizing sub-Lorentzian Heisenberg geodesics are lifts of hyperbolae, with broken lightlike behavior only on the boundary of the timelike region (Borza et al., 8 Sep 2025).

A complete optimal synthesis has now been established. The exponential map

$$X=\partial_x-\frac{y}{2}\partial_z,\qquad Y=\partial_y+\frac{x}{2}\partial_z,\qquad T=\partial_z,$$3

is a real-analytic diffeomorphism, and every timelike future-directed curve

$$X=\partial_x-\frac{y}{2}\partial_z,\qquad Y=\partial_y+\frac{x}{2}\partial_z,\qquad T=\partial_z,$$4

is a length-maximizing geodesic for all $$X=\partial_x-\frac{y}{2}\partial_z,\qquad Y=\partial_y+\frac{x}{2}\partial_z,\qquad T=\partial_z,$$5. For each $$X=\partial_x-\frac{y}{2}\partial_z,\qquad Y=\partial_y+\frac{x}{2}\partial_z,\qquad T=\partial_z,$$6, there is a unique strictly normal timelike maximizer, up to reparametrization. Boundary points in $$X=\partial_x-\frac{y}{2}\partial_z,\qquad Y=\partial_y+\frac{x}{2}\partial_z,\qquad T=\partial_z,$$7 are reached by strictly abnormal broken lightlike maximizers. In this sense, every causally related pair is connected by a maximizing causal geodesic, and the timelike class is non-branching (Borza et al., 3 Apr 2025, Sachkov et al., 2022).

The cut locus is empty: every locally maximizing geodesic is globally maximizing, and there is no cut time. The same global behavior is reflected in the absence of conjugate points along timelike geodesics and in the fact that the exponential map is globally invertible on the timelike domain. Earlier work on the Heisenberg and quaternionic $$X=\partial_x-\frac{y}{2}\partial_z,\qquad Y=\partial_y+\frac{x}{2}\partial_z,\qquad T=\partial_z,$$8-type settings already exhibited explicit normal geodesics and causal reachable sets, and it related the horizontal equations

$$X=\partial_x-\frac{y}{2}\partial_z,\qquad Y=\partial_y+\frac{x}{2}\partial_z,\qquad T=\partial_z,$$9

to the motion of a relativistic particle in a constant homogeneous electromagnetic field. In the Heisenberg case, the planar dynamics is a hyperbolic “boost-like” rotation of the velocity, with the vertical coordinate recovered from the contact constraint (Korolko et al., 2010).

4. Lorentzian length-space structure, Hausdorff dimension, and synthetic curvature failure

Equipped with the sub-Riemannian background distance $[X,Y]=T.$0, the causal relations $[X,Y]=T.$1, and the time-separation $[X,Y]=T.$2, the sub-Lorentzian Heisenberg group becomes a Lorentzian pre-length space and, in fact, a Lorentzian length space. The geometry is strongly causal, causally closed, compatible with the sub-Riemannian distance, non-totally imprisoning, and globally hyperbolic; more precisely, it is $[X,Y]=T.$3-globally hyperbolic. Any two causally related points are joined by a causal geodesic, and $[X,Y]=T.$4 is finite and continuous. For any future-directed causal horizontal curve $[X,Y]=T.$5,

$[X,Y]=T.$6

with the infimum taken over all partitions, which is the Lorentzian length-space analogue of metric length reconstruction (Borza et al., 3 Apr 2025, Borza et al., 8 Sep 2025).

A central result is the Lorentzian Hausdorff-dimension computation. Using the McCann–Fathi–Figalli Lorentzian Hausdorff pre-measure built from causal diamonds, one proves that

$[X,Y]=T.$7

and that the corresponding Lorentzian Hausdorff measure $[X,Y]=T.$8 coincides with Haar measure, that is, Lebesgue measure, up to a constant. The proof combines left-invariance, homogeneity under Heisenberg dilations, explicit control of diamond volumes, and a Lorentzian Ball–Box analogue. Specifically, there exists $[X,Y]=T.$9 such that

$D=\mathrm{span}\{X,Y\}\subset T(H^1),$0

where

$D=\mathrm{span}\{X,Y\}\subset T(H^1),$1

Conversely, there exist $D=\mathrm{span}\{X,Y\}\subset T(H^1),$2 such that for every $D=\mathrm{span}\{X,Y\}\subset T(H^1),$3 one can find $D=\mathrm{span}\{X,Y\}\subset T(H^1),$4 with

$D=\mathrm{span}\{X,Y\}\subset T(H^1),$5

These estimates identify the same anisotropic scaling $D=\mathrm{span}\{X,Y\}\subset T(H^1),$6 that governs the sub-Riemannian Heisenberg group, but now in a timelike Lorentzian-diamond formulation (Borza et al., 8 Sep 2025).

The same paper proves a sharp negative result for synthetic Lorentzian curvature. The sub-Lorentzian Heisenberg group satisfies neither the timelike curvature-dimension condition $D=\mathrm{span}\{X,Y\}\subset T(H^1),$7 for any $D=\mathrm{span}\{X,Y\}\subset T(H^1),$8, $D=\mathrm{span}\{X,Y\}\subset T(H^1),$9, nor the timelike measure contraction property $\mathcal H=\ker\theta$0 for any finite $\mathcal H=\ker\theta$1. The mechanism of failure is explicit. After reduction to $\mathcal H=\ker\theta$2 and Lebesgue measure, a timelike Brunn–Minkowski consequence of $\mathcal H=\ker\theta$3 is contradicted by a midpoint-map Jacobian computation: the determinant of the midpoint map at $\mathcal H=\ker\theta$4 equals $\mathcal H=\ker\theta$5, giving a density drop strictly less than $\mathcal H=\ker\theta$6. Likewise, $\mathcal H=\ker\theta$7 would force a uniform Jacobian lower bound along geodesic transport to a point, but the ratio

$\mathcal H=\ker\theta$8

tends to $\mathcal H=\ker\theta$9 for suitable sequences of initial data with parameter $4$00. The key Jacobian identity is

$4$01

smoothly extended at $4$02 by

$4$03

This stands in sharp contrast with the sub-Riemannian Heisenberg group, where the metric Hausdorff dimension is also $4$04, but Juillet’s result gives $4$05 precisely for $4$06 and $4$07 (Borza et al., 8 Sep 2025).

A frequent misconception is therefore that the sub-Lorentzian Heisenberg group should inherit synthetic curvature properties from its sub-Riemannian counterpart because both have the same homogeneous dimension and the same anisotropic scaling. The available results show the opposite: the timelike transport Jacobians can become arbitrarily small, and the required entropy or distortion convexity inequalities fail.

5. Lorentzian optimal transport and the sub-Lorentzian Monge–Ampère equation

For $4$08, the transport cost is

$4$09

A forward transport map $4$10 must satisfy $4$11 $4$12-almost everywhere, and the forward Monge problem seeks to maximize

$4$13

The Kantorovich relaxation considers causal or timelike couplings

$4$14

and the associated Lorentz-Wasserstein quantity

$4$15

This functional satisfies a reverse triangle inequality,

$4$16

and the optimality theory uses $4$17-cyclical monotonicity, $4$18-concavity, and a Lorentzian form of Kantorovich duality (Borza et al., 3 Apr 2025).

Under absolute continuity, compact support, and timelike support assumptions

$4$19

a Brenier-type theorem holds: there is a unique forward optimal transport map $4$20-almost everywhere, and it is represented through a $4$21-concave potential $4$22 and the sub-Lorentzian exponential map. The same is true for the backward problem, using $4$23. The forward and backward maps are inverses almost everywhere. The Hamiltonian quantity entering the representation is

$4$24

which is the natural sub-Lorentzian co-metric expression. The interpolants

$4$25

parametrize the unique maximizing geodesic from $4$26 to $4$27, and $4$28 is the unique $4$29-geodesic in $4$30 (Borza et al., 3 Apr 2025).

The Jacobian formulation of the sub-Lorentzian Monge–Ampère equation is a change-of-variables identity rather than an elliptic Hessian-determinant equation. If $4$31, then for $4$32-almost every $4$33 and all $4$34,

$4$35

If $4$36 is absolutely continuous, the same relation holds at $4$37. The regularity needed for this identity comes from the semiconcavity of the transport potential and the differentiability and injectivity of $4$38 almost everywhere.

The theory also supplies concrete examples. If $4$39 is the projection $4$40, then a Minkowski forward optimal map $4$41 on the plane lifts to a forward optimal map on $4$42 through left translation and the exponential map, with the identity

$4$43

Right-translations are optimal only in a rigid case: $4$44 is a forward optimal transport map precisely when $4$45 and the projected translation is future-directed in the Minkowski sense almost everywhere. This expresses how causal admissibility and the Heisenberg vertical defect constrain even the simplest symmetry-generated transports (Borza et al., 3 Apr 2025).

6. CMC surfaces, Lorentzian deformations, and broader context

The sub-Lorentzian Heisenberg group has recently become a model space for horizontal area and constant horizontal mean curvature. Its smooth isometry group is

$4$46

with boosts

$4$47

acting as the natural Lorentzian analogue of planar rotations. For a $4$48 embedded surface $4$49, the characteristic set is

$4$50

and on $4$51 there is a horizontal unit normal $4$52. The horizontal mean curvature satisfies the divergence formula

$4$53

and the first variation of horizontal area under suitable compactly supported horizontal variations is

$4$54

Under volume-preserving radial cone variations, area stationarity implies that $4$55 is constant away from the characteristic set (Borza et al., 29 May 2026).

Boost-symmetric constant mean curvature surfaces are classified explicitly. Their construction reduces to constant-curvature profile curves in the Minkowski plane, whose solutions are straight lines when the signed curvature is $4$56 and hyperbolae when it is positive. This produces two main families: the nonzero-curvature family $4$57 and the zero-curvature maximal family $4$58. Within the nonzero-curvature class, the timelike-parameter family with $4$59 is singled out: these surfaces are smooth and acausal, and, up to time-preserving boost, $4$60 can be written as a two-sheeted graph over

$4$61

with explicit $4$62-profile. The same work shows that pseudo-spheres

$4$63

are not constant-mean-curvature surfaces, since their horizontal mean curvature

$4$64

is not constant on any smooth patch. This is a decisive distinction from naive analogies with Lorentzian distance spheres (Borza et al., 29 May 2026).

The current isoperimetric conjecture in this setting is that sub-Lorentzian isoperimetric maximizers belong to the timelike-parameter family $4$65 with $4$66 and $4$67, equivalently, up to a time-preserving boost, to the family $4$68. The paper presents this as the sub-Lorentzian analogue of the Pansu bubbles from sub-Riemannian Heisenberg geometry (Borza et al., 29 May 2026).

A parallel line of work studies full Lorentzian structures on the Heisenberg group and their degeneration to the sub-Lorentzian model. In the family

$4$69

the future cones, attainable sets, exponential maps, and spheres converge as $4$70 to their sub-Lorentzian counterparts. The limiting reachable set is

$4$71

and the limiting exponential map is precisely the sub-Lorentzian one. By contrast, in the family

$4$72

the commutant direction lies inside the union of future and past cones; the system becomes globally controllable, periodic nonspacelike trajectories appear, and there are no length maximizers. The paper abstracts a dichotomy: if the commutant direction is outside the future cone, one has length maximizers between causally related points, whereas if it lies strictly inside the cone union, one has periodic nonspacelike trajectories, global controllability, and absence of maximizers (Sachkov, 24 Nov 2025).

This broader Lorentzian context clarifies a final point. Not every Lorentzian problem on the Heisenberg group is sub-Lorentzian. Some control problems use all three directions $4$73 and indefinite metrics on the full tangent bundle; they are structurally related but geometrically distinct. The sub-Lorentzian Heisenberg group is the horizontal, bracket-generated, causal model in which the vertical direction is produced only indirectly, and its geometry is shaped precisely by that nonholonomic Lorentzian constraint (Galyaev et al., 2024, Sachkov, 24 Nov 2025).